Curved-domain geometry approximation errors

Let $\widehat{\mathrm{\Omega }}\subset {\mathbb{R}}^d$ be a bounded reference domain and let $\mathrm{\Phi },{\mathrm{\Phi }}_h:\overline{\widehat{\mathrm{\Omega }}}\to {\mathbb{R}}^d$ be cellwise $C^2$ admissible maps with matching traces, positive Jacobians, and common image topology. The exact and approximate domains are $\mathrm{\Omega }=\mathrm{\Phi }(\widehat{\mathrm{\Omega }})$ and ${\mathrm{\Omega }}_h={\mathrm{\Phi }}_h(\widehat{\mathrm{\Omega }})$. Define $e_0=\mathrm{\parallel }\mathrm{\Phi }-{\mathrm{\Phi }}_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\widehat{\mathrm{\Omega }})}$ and $e_1=\mathrm{\parallel }D\mathrm{\Phi }-D{\mathrm{\Phi }}_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\widehat{\mathrm{\Omega }};op)}$. A geometry approximation claim must separately state bounds on $e_0,e_1$, both Jacobian lower bounds, and boundary-trace compatibility; none follows from the word “isoparametric.”

The transformed Laplace tensors associated with these maps are $B=J_{\mathrm{\Phi }}D{\mathrm{\Phi }}^{-1}D{\mathrm{\Phi }}^{-T}$ and $B_h=J_{{\mathrm{\Phi }}_h}D{\mathrm{\Phi }}_h^{-1}D{\mathrm{\Phi }}_h^{-T}$ wherever the maps are differentiable. Boundary lifting error and data error are separate quantities and are not included in $e_0$ or $e_1$.